A generalization of Anderson's theorem on unimodal functions

Author:
Somesh Das Gupta

Journal:
Proc. Amer. Math. Soc. **60** (1976), 85-91

MSC:
Primary 26A87; Secondary 52A40

DOI:
https://doi.org/10.1090/S0002-9939-1976-0425050-7

MathSciNet review:
0425050

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Abstract | References | Similar Articles | Additional Information

Abstract: Anderson (1955) gave a definition of a unimodal function on and obtained an inequality for integrals of a symmetric unimodal function over translates of a symmetric convex set. Anderson's assumptions, especially the role of unimodality, are critically examined and generalizations of his inequality are obtained in different directions. It is shown that a marginal function of a unimodal function (even if it is symmetric) need not be unimodal.

**[T]**W. Anderson (1955),*The integral of a symmetric unimodal function over a symmetric convex set and some probability inequalities*, Proc. Amer. Math. Soc.**6**, 170-176. MR**16**, 1005. MR**0069229 (16:1005a)****[G]**S. Mudholkar (1966),*The integral of an invariant unimodal function over an invariant convex set--An inequality and applications*, Proc. Amer. Math. Soc.**17**, 1327-1333. MR**34**#7741. MR**0207928 (34:7741)****[S]**Sherman (1955),*A theorem on convex sets with applications*, Ann. Math. Statist.**26**, 763-767. MR**17**, 655. MR**0074845 (17:655l)**

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1976-0425050-7

Keywords:
Unimodal function,
convex set,
invariance,
marginal function,
inequalities

Article copyright:
© Copyright 1976
American Mathematical Society